A213616 Triangle read by rows, coefficients of the Bernoulli nabla polynomials BN_{n}(x) times A144845(n) in descending order of powers.

1, 2, -3, 6, -18, 13, 2, -9, 13, -6, 30, -180, 390, -360, 119, 6, -45, 130, -180, 119, -30, 42, -378, 1365, -2520, 2499, -1260, 253, 6, -63, 273, -630, 833, -630, 253, -42, 30, -360, 1820, -5040, 8330, -8400, 5060, -1680, 239, 10, -135, 780, -2520, 4998, -6300

Offset

0,2

Comments

The polynomials BN_{n}(x) have the e.g.f. t*exp(t*(x-1))/(exp(t)-1). The adjunct 'nabla' in the name refers to the backward difference operation.

BN_{n}(1) are the Bernoulli numbers.

In the difference table of the Bernoulli polynomials the polynomials BN_{n}(x) appear as the top row (see the link).

Links

Table of n, a(n) for n=0..50.

Peter Luschny, The Computation and Asymptotics of the Bernoulli numbers.

Formula

T(n,k) = A144845(n)*[x^(n-k)]BN{n}(x).

Example

bn(0,x) =  1,
bn(1,x) =  2*x   -   3,
bn(2,x) =  6*x^2 -  18*x   +  13,
bn(3,x) =  2*x^3 -   9*x^2 +  13*x   -   6,
bn(4,x) = 30*x^4 - 180*x^3 + 390*x^2 - 360*x   + 119,
bn(5,x) =  6*x^5 -  45*x^4 + 130*x^3 - 180*x^2 + 119*x - 30.

Maple

seq(seq(coeff(denom(bernoulli(i, x))*bernoulli(i, x - 1), x, i - j), j=0..i), i=0..12);

Mathematica

Table[If[i == 0, 1, 1/First[ FactorTerms[ BernoulliB[i, x]]]]*Table[ Coefficient[ Denominator[ BernoulliB[i, x]]*BernoulliB[i, x-1], x, i-j], {j, 0, i}], {i, 0, 12}] // Flatten (* Jean-François Alcover, Sep 27 2013, after Maple *)

Crossrefs

A053383, A144845, A213615.

Sequence in context: A321399 A169974 A003183 * A359180 A131788 A294455

Adjacent sequences: A213613 A213614 A213615 * A213617 A213618 A213619

Keyword

sign,tabl

Author

Peter Luschny, Jun 16 2012

Status

approved